Trace map and regularity of finite extensions of a DVR

Journal of Number Theory 172 (2017) 133–144

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Journal of Number Theory www.elsevier.com/locate/jnt

Trace map and regularity of finite extensions of a DVR Fabio Tonini Freie Universität Berlin, FB Mathematik und Informatik, Arnimallee 3, Zimmer 112A, 14195 Berlin, Germany

a r t i c l e

i n f o

Article history: Received 10 May 2016 Received in revised form 20 August 2016 Accepted 22 August 2016 Available online 8 October 2016 Communicated by D. Goss

a b s t r a c t We interpret the regularity of a finite and flat extension of a discrete valuation ring in terms of the trace map of the extension. © 2016 Elsevier Inc. All rights reserved.

Keywords: Regularity Trace map Ramification index Tame

1. Introduction Let R be a ring and A be an R-algebra which is projective and finitely generated as  A/R : A −→ A∨ = R-module. We denote by trA/R : A −→ R the trace map and by tr HomR (A, R) the map a −→ trA/R (a · −). It is a well known result of commutative algebra that the étaleness of the extension A/R is entirely encoded in the trace map  A/R : A −→ A∨ is an isomorphism (see trA/R : A/R is étale if and only if the map tr [Gro71, Proposition 4.10]). In this case, if R is a DVR (discrete valuation ring) it follows E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jnt.2016.08.019 0022-314X/© 2016 Elsevier Inc. All rights reserved.

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that A is regular (that is a product of Dedekind domains), while the converse is clearly not true because extensions of Dedekind domains are often ramified. In this paper we show how to read the regularity of A in terms of the trace map trA/R . In order to express our result we need some notations and definitions. Let us assume from now on that the ring R is a DVR with residue field kR . We first extend the notion of tame extensions and ramification index: given a maximal ideal p of A we set

e(p, A/R) =

dimkR (Ap ⊗R kR ) [k(p) : kR ]

where k(p) = A/p, and we call it the ramification index of p in the extension A/R. Notice that e(p, A/R) ∈ N (see Lemma 2.3). We say that A/R is tame (over the maximal ideal of R) if the ramification indexes of all maximal ideals of A are coprime with char kR . Those definitions agree with the usual ones when A is a Dedekind domain. We also set  A/R tr

QA/R = Coker(A −→ A∨ ), f A/R = l(QA/R ) where l denotes the length function. Alternatively f A/R can be seen as the valuation of  A/R . We also denote by | Spec(A ⊗R kR )| the number of the discriminant section det tr primes of A ⊗R kR : this number can also be computed as 

| Spec(A ⊗R kR )| =

[Fp : kR ]

p maximal ideals of A

where Fp denotes the maximal separable extension of kR inside k(p) = A/p (see Corollary 2.4). Finally we will say that A/R has separable residue fields (over the maximal ideal of R) if for all maximal ideals p of A the finite extension k(p)/kR is separable. The theorem we are going to prove is the following: Main theorem. Let R be a DVR and A be a finite and flat R-algebra. Then we have the inequality f A/R ≥ rk A − | Spec(A ⊗R kR )| and the following conditions are equivalent: 1) the equality holds in the inequality above; 2) A is regular and A/R is tame with separable residue fields; 3) A/R is tame with separable residue fields and the R-module QA/R is defined over kR , that is mR QA/R = 0, where mR denotes the maximal ideal of R.

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The implication 2) =⇒ 1) is classical, because it follows directly from the relation between the different and ramification indexes (see [Ser79, III, §6, Proposition 13]). Notice that the étaleness of A/R is equivalent to any of the following conditions: f A/R = 0, rk A = | Spec(A ⊗R kR )|, QA/R = 0. The above result has already been proved in my Ph.D. thesis [Ton13, Theorem 4.4.4] under the assumption that a finite solvable group G acts on A in a way that AG = R and using a completely different strategy: using induction and finding a filtration of R-algebras of R ⊆ A starting from a filtration of normal subgroups of G. Main theorem is an essential ingredient in the proof of [Ton15, Theorem C] which generalizes [Ton13, Theorem 4.4.7]. The paper is organized as follows. In the first section we discuss the notion of tameness and ramification index and recall some basic facts of commutative algebra and, in particular, about the trace map. In the second section we prove the Main theorem. 1.1. Notation All rings and algebras in this paper are commutative with unity. Given a ring R and a prime q we denote by k(q) the residue field of Rq . If A is a finite R-algebra we say that A/R has separable residue fields over a prime q of R if for all primes p of A lying over q the finite extension of fields k(p)/k(q) is separable. We say that A/R has separable residue fields if it has this property over all primes of R. The following conditions are equivalent (see [Gro64, Proposition 1.4.7]): A/R is finite, flat and finitely presented; A is finitely generated and projective as R-module; A is finitely presented as R-module and for all primes q of R the Rq -module Aq is free. In this situation there is a well-defined trace function HomR (A, A) −→ R which extends the usual trace of matrices and commutes with arbitrary extensions of scalars. The trace map of A/R, denoted by trA/R : A −→ R (or simply trA ), is the composition A −→ HomR (A, A) −→ R, where the first map is induced by the multiplication in A. If R is a local ring we denote by mR its maximal ideal and by kR its residue field. A DVR is a discrete valuation ring. If k is a field we denote by k an algebraic closure of k and by ks a separable closure of k. 2. Preliminaries on tameness and trace map We fix a ring R and a finite, flat and finitely presented R-algebra A over R, that is an R-algebra A which is finitely presented as R-module and such that Aq is a free Rq -module of finite rank for all prime q of R. Definition 2.1. Given a prime ideal p of A lying above the prime q of R we set e(p, A/R) =

dimk(q) (Ap ⊗Rq k(q)) [k(p) : k(q)]

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and we call it the ramification index of p in the extension A/R. We say that A/R is tame in p ∈ Spec A if e(p, A/R) is coprime with char k(q), is tame over q ∈ Spec R if it is tame over all primes of A over q and, finally, we say that it is tame if it is tame over all primes of R (or in all primes of A). Remark 2.2. The number e(p, A/R) is always a natural number as shown below. Moreover if R and A are Dedekind domains, the notion of ramification index agrees with the usual one. Lemma 2.3. Let p be a prime of A lying over the prime q of R and denote by F the maximal separable extension of k(q) inside k(p). Then e(p, A/R) is a natural number and Ap ⊗Rq k(q) B1 × · · · × Bs with Bi local, dimk(q) Bi = e(p, A/R)[k(p) : F ] and s = [F : k(q)] Proof. We can assume that R = k(q) = k is a field and that A is local. Set also L = k(p) and write L ⊗k k C1 × · · · × Cr , A ⊗k k B1 × · · · × Bs for the decompositions into local rings. In particular we have that r = s because A ⊗k k −→ L ⊗k k is surjective with nilpotent kernel. Moreover this map splits as a product of surjective maps Bi −→ Ci . Denote by Ji their kernels and by φ : A ⊗k k −→ B1 ×· · ·×Bs the previous isomorphism of rings. For all t ∈ N (including t = 0) we have φ((mA ⊗k k)t ) = J1t × · · · × Jst In particular for all t ∈ N we have 

J1t J1t+1



 × ··· ×

Jst Jst+1



(mA ⊗k k)t mtA ⊗k k mt+1 (mA ⊗k k)t+1 A f (t)

(L ⊗k k)f (t) C1

× · · · × Csf (t)

as L ⊗k k-modules, where f (t) = dimL (mtA /mt+1 A ). We can conclude that 

Jit Jit+1

 f (t)

Ci

for all i, t

In particular dimk Bi =

 t∈N

 dimk

Jit Jit+1

 = (dimk Ci )

 t∈N

f (t)

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because Ji is nilpotent. Similarly we have dimk A =

 t∈N

 dimk

mtA mt+1 A

 = [L : k]



f (t)

t∈N

In particular [L : k] | dimk A, so that the ramification index is a natural number. By a direct computation we see that everything follows if we show that dimk Ci = [L : F ]. Since F/k is separable we know that F ⊗k k k embedding σ : F −→ k such that σ|k = idk and

[F :k]

. Each factor corresponds to an



L ⊗k k L ⊗F (F ⊗k k)

L ⊗F,σ k

σ∈Homk (F,k)

This is exactly the decomposition into local rings because, since L/F is purely inseparable, all the rings L ⊗F,σ k are local. This ends the proof. 2 Corollary 2.4. Let q be a prime of R. Then | Spec(A ⊗R k(q))| =



[Fp : k(q)]

p primes of A over q

where Fp denotes the maximal separable extension of k(q) inside k(p). Proof. It follows from Lemma 2.3 using the fact that A ⊗R k(q) is the product of the Ap ⊗Rq k(q) for p running through all primes of A over q. 2 Definition 2.5. Let p be a prime of A lying over the prime q of R. We denote by h(p, A/R) the common length of the localizations of Ap ⊗Rq k(q), that is, following notation from Lemma 2.3, h(p, A/R) = dimk Bi = e(p, A/R)[k(p) : F ]. Lemma 2.6. Let p be a prime of A, R be an R-algebra and p be a prime of A = A ⊗R R over the prime p. Then h(p, A/R) = h(p , A /R ) Proof. We can assume that R = k and R = k are fields and that A is local. Moreover by definition of the function h(−) we can also assume that k and k are algebraically closed. In this case h(p, A/k) = dimk A and, since A = A ⊗k k is again local, h(p , A /k ) = dimk A = dimk A. 2 Corollary 2.7. Let p be a prime of A. Then A/R is tame in p and k(p)/k(q) is separable if and only if the number h(p, A/R) is coprime with char k(q).

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Proof. We can assume that R = k = k(q) is a field and A is local with residue field L. Let also F be the maximal separable extension of k inside L. Thanks to Lemma 2.3, the last condition in the statement is that the number e(mA , A/k)[L : F ] is coprime with char k. Since L/F is purely inseparable, so that [L : F ] is either 1 or a power of char k, this is the same as A/k being tame and L = F , that is L/k is separable. 2 Remark 2.8. Let q be a prime of R, R be an R-algebra and q  be a prime of R over R. By Lemma 2.6 and Corollary 2.7 it follows that A/R is tame over q and A ⊗R k(q) has separable residue fields if and only if the same is true for the extension (A ⊗R R )/R with respect to the prime q  . On the other hand tameness alone does not satisfy the same base change property, and, in particular, the function e(−) does not satisfy Lemma 2.6. The counterexample is a finite purely inseparable extension L/k: we have that e(mk , L/k) = 1, so that L/k is tame, while e(mk , L ⊗k k/k) = [L : k] because L ⊗k k is local, so that L ⊗k k/k is not tame. Lemma 2.9. Assume that R = k is a field, that A is local and set π : A −→ kA for the projection. Then trA/k = e(mA , A/k) trkA /k ◦π Proof. Set P = mA , L = kA and let x1,i , . . . , xri ,i ∈ P i be elements whose projections form an L-basis of P i /P i+1 . We set x1,0 = 1. Let also y1 , . . . , ys ∈ A be elements whose projections form a k-basis of L, where s = dimk L. It is easy to see that the collection {xα,i yβ }1≤α≤ri ,1≤β≤s is a k-basis of P i /P i+1 for all i ≥ 0. By an inverse induction starting from the nilpotent index of P , it also follows that Bn = {xα,i yβ }1≤α≤ri ,1≤β≤s,i≥n is a k-basis of P n for all n ≥ 0. In particular, when n = 0 we get a k-basis of A = P 0 . We are going to compute the trace map trA over the basis B0 . Consider an index i > 0. For all possible α, β, γ, δ, j we have that z = (xα,i yβ )(xγ,j yδ ) ∈ P i+j ⊆ P j+1 Thus z is a linear combination of vectors in Bj+1 , which does not contain (xγ,j yδ ). It follows that trA (xα,i yβ ) = 0 for all i > 0, that is trA (P ) = 0 which agrees with the formula in the statement. It remains to compute trA (yβ ). Write yβ yδ =

 q

bβ,δ,q yq + uβ,δ with uβ,δ ∈ P and bβ,δ,q ∈ k

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It follows that trL/k (π(yβ )) =



bβ,δ,δ

δ

Let’s multiply now yβ with an element xα,i yδ , obtaining z = yβ (xα,i yδ ) = xα,i uβ,δ +



bβ,δ,q xα,i yq

q

If i = 0, so that α = 1 and x1,0 = 1, the coefficient of z with respect to yδ is bβ,δ,δ because uβ,δ ∈ P . If i > 0 then the coefficient of z with respect to (xα,i yδ ) is again bβ,δ,δ because xα,i uβ,δ ∈ P i+1 and thus can be written using only the vectors in Bi+1 . In conclusion we have that trA (yβ ) =



   bβ,δ,δ = ( bβ,δ,δ )( 1) = trL (π(yβ ))C with C = ( 1)

i,α,δ

α,i

δ

i,α

Thus trA = C trL ◦π and, finally, C=(



1) =

i,α

 i≥0

dimL (

Pi dimk A = e(mA , A/k) )= i+1 P [L : k]

2

Corollary 2.10. Assume that R and A are local. Then 1) trA/R (mA ) ⊆ mR ; 2) if kA = kR and rk A ∈ R∗ then Ker trA/R ⊆ mA . Proof. Since trA/R ⊗R kR = tr(A⊗R kR )/kR point 1) follows from Lemma 2.9 because tr(A⊗R kR )/kR (mA⊗R kR ) = 0. Assume now the hypothesis of 2) and let x ∈ Ker trA . If x∈ / mA there exists λ ∈ R∗ such that y = x − λ ∈ mA , so that trA (x) = 0 = rk Aλ + trA (y) ∈ R∗ + mR = R∗ which is impossible.

2

Lemma 2.11. If R and A are local then trA/R : A −→ R is surjective ⇐⇒ h(mA , A/R) and char kR are coprime Proof. By Nakayama’s lemma trA/R : A −→ R is surjective if and only if trA/R ⊗R kR = trA⊗kR /kR : A ⊗ kR −→ kR is so. Thus we can assume that R = k is a field. By Lemma 2.9 trA/k is surjective if and only if trkA /k is surjective, i.e. kA /k is separable, and e(mA , A/k) ∈ k∗ . The result then follows from Corollary 2.7. 2

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3. Regularity of finite extensions of DVR We fix a DVR R and a finite and flat R-algebra A, so that A is free of finite rank rk A as R-module. We will use the following notation  A/R : A −→ A∨ , a −→ trA/R (a · −) tr  A/R tr

QA/R = Coker(A −→ A∨ ), f A/R = l(QA/R ) where l denotes the length function. For simplicity we will replace A/R with A if no confusion can arise. Remark 3.1. By standard arguments we have that f A coincides with the valuation over  A ). Moreover the following conditions are equivalent: R of det(tr 1) A is generically étale over R; 2) f A < ∞; 3) trA : A −→ A∨ is injective. In particular we see that all three conditions in Main theorem implies that A/R is generically étale. This also means that A/R is tame with separable residue fields if and only if A ⊗R kR /kR , or A ⊗R kR /kR , is so.  A : A −→ A∨ with respect If β = {x0 , . . . , xn } is an R-basis of A then the matrix of tr to β and its dual is given by T = (trA (xi xj ))i,j . In particular we can conclude that, if trA : A −→ R is not surjective, then f A ≥ rk A, because all entries of T belongs to mR . If trA (1) = rk A ∈ R∗ we have a decomposition A = R1 ⊕ Ker trA . In particular if x0 = 1 and x1 , . . . , xn ∈ Ker trA (such a basis always exists locally) the matrix T has the form 

 rk A 0

0 N

and therefore QA Coker N . Remark 3.2. Let Rs be the strict Henselization of R and set As = A ⊗R Rs . In particular s Rs is a discrete valuation ring with residue field the separable closure kR of kR . Using s that the extension R −→ R is faithfully flat and unramified, it is easy to see that s

f A/R = f A

/Rs

, | Spec(A ⊗R kR )| = | Spec(As ⊗Rs kRs )|

that QA/R is defined over kR if and only if QAs /Rs is defined over kRs and that A/R is tame with separable residue fields if and only if As /Rs is so. Since Rs is Henselian, we have a decomposition As = A1 × · · · × Aq where the Aj are local rings finite and flat over Rs . Moreover Aj ⊗Rs kRs are still local, so that q =

F. Tonini / Journal of Number Theory 172 (2017) 133–144

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 As /Rs : As −→ (As )∨ is the direct sum of the tr  Aj /Rs : Aj −→ | Spec(A ⊗R kR )|. Since tr ∨ (Aj ) we have QAs /Rs



s

QAj /Rs and f A

/Rs

=

j



f Aj /Rs

j

Finally we have that the following three conditions are equivalent: A is regular; As is regular; Aj is regular for all j = 1, . . . , q. Proof. (of Main theorem) By Remark 3.1 and Remark 3.2 we can assume that R is strictly Henselian, that A is local, so that | Spec(A ⊗R kR )| = 1, and that A/R is generically étale. Moreover in this case the following three conditions are equivalent by Corollary 2.7 and Lemma 2.11: A/R is tame with separable residue fields; kA = kR and rk A ∈ R∗ ; trA/R : A −→ R is surjective. Inequality and 1) ⇐⇒ 3). By Remark 3.1 we can assume that trA : A −→ R is surjective, so that kA = kR , rk A ∈ R∗ . By Corollary 2.10 we also have Ker trA ⊆ mA and trA (mA ) ⊆ mR . Using Remark 3.1 and its notation, we see that f A = vR (det N ). If i, j ≥ 1 then xi xj ∈ mA and thus trA (xi xj ) ∈ mR , that is all entries of N belongs to mR . If π ∈ mR is an uniformizer of R we can write N = πN  where N  is a matrix with entries in R. In particular det N = π rk A−1 det N  Applying the valuation of R we get f A = vR (det N ) = rk A − 1 + vR (det N  ) and the desired inequality. rk A−1 If QA Coker(N ) is defined over kR we obtain a surjective map kR −→ QA ⊗ A A kR QA and therefore that f = l(QA ) ≤ rk A − 1. Conversely, if f = rk A − 1, which means that N  : Rrk A−1 −→ Rrk A−1 is an isomorphism, we have QA Coker(πN  ), so rk A−1 that QA kR is defined over kR as required. 2) =⇒ 1) Let t ∈ A be a generator of the maximal ideal. The kR -algebra A ⊗ kR is local, with residue field kR and its maximal ideal is generated by the projection t of t. It is easy to conclude that A ⊗ kR = kR [X](X N ) where N = rk A and X corresponds to t. By Nakayama’s lemma it follows that 1, t, . . . , tN −1 is an R-basis of A and thus that A R[Y ]/(Y N − g(Y )) where Y corresponds to t, deg g < N and all coefficients of g are in mR . Since mA = t, mR A , the condition that A is regular tells us that vR (g(0)) = 1.

A : A −→ A∨ writing We are going to compute the valuation of the determinant of tr this map in terms of the basis 1, t, . . . , tN −1 , that is the valuation of the determinant of the matrix (trA (ti+j ))0≤i,j 0. In particular, since vR (g(0)) = 1, we also have vR (qN ) = 1. Moreover it follows by induction that vR (qs ) > 1 if s > N . Set SN for the group of permutations of the set {0, 1, . . . , N − 1}. We have det((trA (ti+j ))0≤i,j
 σ∈SN

(−1)sgn(σ) zσ where zσ = q0+σ(0) q1+σ(1) · · · qN −1+σ(N −1)

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We claim that vR (zσ ) ≥ N for all σ ∈ Sn but the permutation σ(0) = 0 and σ(i) = N − i for i = 0, for which vR (zσ ) = N −1. This will conclude the proof. Let σ ∈ SN . If σ(0) = 0, then all the N -factors of zσ are in mR , which implies that vR (zσ ) ≥ N . Thus assume that σ(0) = 0. Since q0 = trA (1) = rk A ∈ R∗ we see that zσ is, up to q0 , the product of N − 1 elements of mR . If one of those factors is of the form qi+σ(i) with i + σ(i) > N then vR (zσ ) ≥ N because vR (qs ) ≥ 2 if s > N . Thus the only case left is when σ(i) ≤ N − i for all 0 < i < N and σ(0) = 0. But, arguing by induction, this σ is unique and it is given by σ(0) = 0 and σ(i) = N − i. In this case we have zσ = rk A(tr(tN ))N −1 and therefore vR (zσ ) = N − 1 as required. 1) =⇒ 2) Since 1) ⇐⇒ 3), we already know that A/R is tame with separable residue fields. We have to prove that A is regular. Set k(R) for the fraction field of R. Since A ⊗ k(R) is etale over k(R), it is a product of fields L1 , . . . , Ls which are separable extensions of k(R). Let B be the integral closure of R inside A ⊗ k(R). We have that A ⊆ B and that B = B1 × · · · × Bs where Bi is the integral closure of R inside Li . Since R is strictly Henselian and the Bi are domains, it follows that they are local. Moreover since R is a DVR we can also conclude that the Bi are DVR. We are going to prove that s = 1 and A = B. Notice that rk B = rk A and denote by j : A −→ B the inclusion. By computing trA and trB over A ⊗ k(R) B ⊗ k(R) we can conclude that (trB )|A = trA . In particular we obtain a commutative diagram of free R-modules

A

 tr A

j∨

j

B

A∨

 tr B

B∨

Notice that det j = det j ∨ . Thus taking determinants and then valuations we obtain the expression f A = 2vR (det j) + f B = 2vR (det j) +

s 

f Bi

i=1

Since k is separably closed and thanks to Corollary 2.4 we have that | Spec(Bi ⊗R k)| = 1. In particular f Bi ≥ rk Bi − 1 by the inequality in the statement. Since f A = rk A − 1 we get s ≥ 1 + 2vR (det j) We are going to prove that vR (det j) ≥ s − 1. This will end the proof because it implies s = 1 and vR (det j) = 0, that is B = A. Since kA = kR we have

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143

vR (det j) = l(B/A) ≥ dimkR (B/(A + mA B)) Denote by e1 , . . . , es ∈ B the idempotents corresponding to the decomposition B = B1 ×· · ·×Bs . We will prove that e2 , . . . , es are kR -linearly independent in B/(A +mA B), that is we prove that if x = x1 e1 + · · · + xs es ∈ A + mA B where x1 = 0 and xi ∈ R then all xi are in the maximal ideal mR . Notice that, since kA = kR , 1 and mA generates A as R-module. In particular A + mA B is generated by 1 and mA B as R-module. Moreover since the maps A −→ B −→ Bi map mA inside mBi it follows that mA B ⊆ mB1 × · · · × mBs . Thus x can be written as x = α + c1 e1 + · · · + cs es with α ∈ R and ci ∈ mBi inside B. In particular 0 = x1 = α + c1 in B1 , which implies that α ∈ mB1 ∩ R = mR . Thus if i > 0 we have xi = α + ci ∈ mBi ∩ R = mR as required. 2 We show via some examples how the conditions of tameness and separability of residue fields in Main theorem cannot be omitted. Example 3.3. Let R = Z2 be the ring of 2-adic numbers and consider the R-algebra A=

R[X] with c ∈ R (X 2 − c)

A : A −→ A∨ is We have that trA (X) = 0 and trA (X 2 ) = 2c, so that the matrix of tr  M=

 2 0 0 2c

In particular f A = vR (det M ) ≥ 2 > rk A − | Spec(A ⊗R kR )| and QA is defined over F2 if and only if c ∈ R∗ . Assume c = 2t + d2 with t, d ∈ R, so that A/mR A F2 [Y ]/(Y 2 ), A is local with maximal ideal (mR , X − d), has separable residue fields and it is not tame. We see that QA is defined over F2 if and only if d ∈ R∗ , while A is regular if and only if t ∈ R∗ . Assume c ∈ R∗ and that c is not a square in F2 . In this case A is local with maximal ideal mR A, A/R is tame, its residue field is not separable, it is regular and QA is defined over F2 .

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Acknowledgments I would like to thank Angelo Vistoli and Dajano Tossici for all the useful conversations we had and all the suggestions I received. References [Gro64] Alexander Grothendieck, EGA4-1 - Étude locale des schémas et des morphismes de schémas (Première partie) - Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné), Publications Mathématiques, vol. 20, Institut des Hautes Études Scientifiques, 1964. [Gro71] Alexander Grothendieck, Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224, Springer-Verlag, jun 1971. [Ser79] Jean-Pierre Serre, Local Fields, Graduate Texts in Mathematics, vol. 67, Springer, New York, 1979. [Ton13] Fabio Tonini, Stacks of ramified Galois covers, PhD thesis, July 2013. [Ton15] Fabio Tonini, Ramified Galois covers via monoidal functors, arXiv:1507.05309, July 2015; Transform. Groups (2016), http://dx.doi.org/10.1007/s00031-016-9395-4, in press.